Similar issues here. Firefox and Chrome got added with no problem, but Safari kept asking me to grant full disk access no matter how many times I did so. I restarted as LEoM suggested. Clicking for Safari access then produced a prolonged spinning beachball, but one that did resolve itself and it seems to be working now.

Also Note:
Cookie looks for the existence of the~/Library/Cookies/Cookies.binaryCookies

If you don’t have the file on your system, Cookie thinks it doesn’t have dull disk access. If you have just installed your Operating System, or have never run Safari in regular NON private browsing mode. It won’t exist, and you will need to open Safari in NON private browsing mode and visit a few websites to generate the file.

The nuclear option still does not resolve the issue: Safari cannot be enabled.
Brave and Opera had no problems, Firefox was helped by the nuclear option.
I mainly use Brave, but airdrop always requires Safari, and some websites require Firefox to be used.

@smolk Cookie checks that it has access to the Safari Cookie file:~/Library/Cookies/Cookies.binarycookies

Also some users have found that if when enabling browsers - NOT selecting the Global permissions option, but selecting specific folder helps. If you initially opted to give Cookie Global access, try this:

That file did not exist, but files with the same name but a temp ending did. Renaming one did not help.
But then I deleted all (app, container files) and reinstalled everything from scratch and noticed the file ~/Library/Cookies/Cookies.binarycookies did not exist.
Thought once, then opened a website in Safari - and the file came into being. I immediately had Safari enabled.
The file is indeed the crux and it is created by actually using Safari.

I continue to struggle with granting Cookie "Full Disk Access". No matter how many times I go through the process, I continue to get the attached error message when I launch the Cookie interface. Please advise. Thank you.

I had this problem after initially upgrading to Cookie6 but it was resolved.

BUT NOW, after allowing AppStore to upgrade me to 6.0.12, I am back in the loop of having to grant access.
System level is macOS 10.15.3
Cookie does have full disk access. I have quit Cookie, removed disk access, restarted system. Doesn't help.
I have had to disable Cookie to get ANY work done.